Extreme magnetoresistance at high-mobility oxide heterointerfaces with dynamic defect tunability

Magnetic field-induced changes in the electrical resistance of materials reveal insights into the fundamental properties governing their electronic and magnetic behavior. Various classes of magnetoresistance have been realized, including giant, colossal, and extraordinary magnetoresistance, each with distinct physical origins. In recent years, extreme magnetoresistance (XMR) has been observed in topological and non-topological materials displaying a non-saturating magnetoresistance reaching 103−108% in magnetic fields up to 60 T. XMR is often intimately linked to a gapless band structure with steep bands and charge compensation. Here, we show that a linear XMR of 80,000% at 15 T and 2 K emerges at the high-mobility interface between the large band-gap oxides γ-Al2O3 and SrTiO3. Despite the chemically and electronically very dissimilar environment, the temperature/field phase diagrams of γ-Al2O3/SrTiO3 bear a striking resemblance to XMR semimetals. By comparing magnetotransport, microscopic current imaging, and momentum-resolved band structures, we conclude that the XMR in γ-Al2O3/SrTiO3 is not strongly linked to the band structure, but arises from weak disorder enforcing a squeezed guiding center motion of electrons. We also present a dynamic XMR self-enhancement through an autonomous redistribution of quasi-mobile oxygen vacancies. Our findings shed new light on XMR and introduce tunability using dynamic defect engineering.

Supplementary Section 6: Annealing   Supplementary Section 8: Magnetoresistive scaling Fitting the magnetoresistance as a function of the mobility reveals that  ∝  0 1.4±0.17for the compiled data with  0 < 20,000 cm 2 /Vs.For data with higher mobility, the trend remains positive but with an imprecise scaling relationship as the data is more scattered in this region.The scattered points originate from sample-to-sample variations as well as from the inclusion of data with   > 8 ⋅ 10 14 cm -2 where bulk conductivity is likely to start emerging.Rs) normalized with the zero-field sheet resistance when these are plotted against the ratio B/Rs.Despite collapsing the curves to a fair extent, the scaling is not perfect.In particular, deviations of Kohler's rule were previously found at temperatures below 40 K for γ-Al2O3/SrTiO3, which was attributed to interactions between itinerant electrons and a magnetic order.This interaction was inferred to cause a negative magnetoresistance contribution to be added to an overall positive magnetoresistance of γ-Al2O3/SrTiO3 (4).

Supplementary Section 9: Kohler scaling
Supplementary Section 10: Origin of the extreme magnetoresistance and its tunability We here turn our attention to the origin of the high magnetoresistance and its linear behavior at high magnetic fields.The most common mechanism for archetypical XMR materials is charge compensation (5).Despite that the universal triangular phase diagram is also observed for γ-Al2O3/SrTiO3, the linear rather than quadratic magnetoresistance combined with a band structure comprising only n-type carriers exclude this mechanism.The ARPES data further show that the unsaturated MR of γ-Al2O3/SrTiO3 also does not come from an open Fermi surface (6) and that heavy bands are present in γ-Al2O3/SrTiO3 rather than the steep bands observed to promote the mobility and magnetoresistance in many XMR semimetals.The linear magnetoresistance is unlikely to be resulting from transport in the extreme quantum limit as it would require a carrier density below (/ℏ) 3/2 = 3 ⋅ 10 17 cm −3 (7,8).This carrier density is similar to the high-mobility SrTiO3 thin films grown with very low doping of SrTiO3 by molecular beam epitaxy (9), but much lower than that obtained in γ-Al2O3/SrTiO3 and generally other SrTiO3-based heterointerfaces grown by pulsed laser deposition (10,11).
In contrast, the linear magnetoresistance is likely to arise from inhomogeneities in the conductivity from either a strongly disordered material in the classical transport regime (12,13) or a weakly disordered material in the semiclassical regime (14).The geometric aspect of the magnetoresistance is consistent with Figure S14, which probes the geometric contribution to the magnetoresistance by comparing magnetotransport with a van der Pauw and linear contact configuration.The degree of disorder can be assessed through the thermal energy required to relax the disorder-induced high-resistive state (region 2 of Figure 1 in the main text).This energy (a few meV, see Figure 2b) is much smaller than the measured Fermi energy level (several tens of meV, see Figure 4e), which classifies the γ-Al2O3/SrTiO3 heterostructures with high MR as a weakly disordered medium.The claim of weak disorder is further supported by the high mobility and the slowly varying magnetic stray field for the heterostructures with high linear magnetoresistance.From the magnetoresistive scaling, the linear magnetoresistance is observed for values of  0  > 10 for the sample presented in Figure 4g, which places the transport in the semiclassical regime.Therefore, the magnetoresistance in γ-Al2O3/SrTiO3 appears to be consistent with the guiding-center model in the semi-classical transport regime with weak disorder.The guiding center model applies to a conducting medium with a 3D or quasi-2D character with a slowly varying disorder potential compared to the cyclotron radius.In addition, a key experimental hallmark is a field-independent Hall angle.As we argue below, these assumptions are all fulfilled in high-mobility γ-Al2O3/SrTiO3.

Spatial extent of the electron gas:
The large, linear magnetoresistance emerges at high sheet carrier densities on the order of   = 5 ⋅ 10 14 cm -2 and can be further enhanced by aging.The high sheet carrier density, the transition in the local current distribution observed by scanning SQUID as well as the ARPES data point towards an extended depth distribution of the itinerant charges.This is supported by the predominant population of heavy 3D bands and the enhancement of the magnetoresistance through aging, which both boost the mobility and are predicted to expand the depth distribution (10,15).The depth distribution of γ-Al2O3/SrTiO3 samples with similar mobility and carrier density has been assessed by angle-dependent Shubnikov-de Haas oscillations (16), angle-dependent X-ray photoemission spectroscopy ( 16), hard x-ray photoemission spectroscopy (17) and infrared ellipsometry (18), which point towards the majority of electrons residing within the first 1-10 nm of the γ-Al2O3/SrTiO3 interface.This indicates that a significant fraction of the carriers resides in an interface-near region of SrTiO3 with confinement formed by oxygen vacancies stabilized at the γ-Al2O3/SrTiO3 interface, whereas the remaining electrons distribute deeper into SrTiO3.This picture is consistent with numerical simulations on γ-Al2O3/SrTiO3 (2, 10).

Slowly varying disorder potential:
The characteristic length () of the disordered potential variations can be estimated by  ≫   = ℏ     where   ,   and    denote the cyclotron radius, Fermi momentum and crossover magnetic field to the linear magnetoresistance, respectively (14).For Fermi momenta of the heavy bands along the heavy and light direction,   are 0.34 Å -1 and 0.069 Å -1 (Figure 4e) yielding cyclotron radii of 750 and 150 nm at    ~3 T at 2 K (Figure 2c), respectively.Hence, the characteristic disorder length is expected to be on the scale of several hundreds of nanometers to micrometers.In contrast to XMR materials with steep bands, the heavy bands and large carrier density of γ-Al2O3/SrTiO3 also translate into a large Fermi momentum, which sets the lower bound for the characteristic disorder length.Combined with a very high dielectric constant, the characteristic disorder length in γ-Al2O3/SrTiO3 is an order of magnitude larger than in the 3D Dirac metals discussed in Ref. (14).The large, linear magnetoresistance emerges at high sheet carrier densities where the magnetic field from the local current distribution has transitioned from a stripe behavior to slowly varying modulations with a characteristic length of a few tens of micrometers (Figure 4a).We stress that the Oersted field may be broadened by the distance between the pick-up loop and the spatially extended current, and that the actual modulation of the current may have finer details not resolvable here.Irrespective of this, the scanning SQUID images justify the assumption of a slowly varying disorder potential.This is further supported by the high dielectric constant of SrTiO3 exceeding 10,000 (19), which favors a slowly varying disorder potential by screening scattering sites and inhomogeneities.The scanning SQUID images (Figure 4a) also reveal that the striped current modulations and magnetic order (4) along the tetragonal domain walls are not a likely origin of disorder and linear magnetoresistance.

Saturating Hall angle:
As depicted in Figure 2c, tan(  ) saturates above a characteristic crossover magnetic field, which coincides with the emergence of linear magnetoresistance.In this case, where the large value of =0.49signifies a strong coupling between   and   −1 .If electronic conduction with  ≫ 1 takes place in a single band or N bands, then   () = / or   () = Σ     /, respectively.In the regime where  is field-independent, the magnetoresistance yields: for conduction in a single band where   (0) = 1/ or using a population weighted mobility  ≈ Σ       /  for multiband conduction.The field-independent  captures the linear scaling of the magnetoresistance when varying the magnetic field and mobility both in single-and multiband systems.Using  ≈ 100,000 cm 2 /Vs and  = 0.49, we obtain a lower value of (15) ≈  = 7350% compared to experiments, which suggests that an additional component influences   (B).The conductivity tensor elements are displayed and discussed further in Supplementary Section 11, where it is shown that   exhibits an inversely proportional field dependence at high magnetic fields, but that it deviates significantly from the conventional conductivity tensor (see Figure S15).As discussed in this supplementary section, the anomalous Hall effect may be a likely candidate for this additional source.

Physical picture of the linear magnetoresistance:
The field-independent  is predicted by the guiding center model (14).For itinerant charge carriers strictly confined to a 2D sheet, the semiclassical transport with  0  > 1 and  ≫   causes the carriers to undergo rapid cyclotron motion while following an overall guiding center motion along the equipotential contours of the disorder potential (Figure S12a).In strict 2D systems, the confinement in the x/y-plane causes the guiding center motion to follow closed orbits.Relaxing the strict 2D confinement results in an interesting case where an overall motion in the x/yplane is enabled by motion in the z-direction, which allows carriers to escape the closed orbits by moving across layers with different disorder landscapes (Figure S12b).If the kinetic energy along the z-direction only allows movement across layers at locations where the variations in the disorder landscape does not exceed the kinetic energy, the current gets squeezed along z.This temporary constraint in the x/y-plane is associated with a slow guiding center velocity, which leads to the linear magnetoresistance (14).In contrast, if the out-of-plane kinetic energy exceeds the disorder potential variations, free movement across layers is observed, resulting in a magnetoresistance departing from linearity (Figure S12c).This situation arises when the thermal energy of the system increases, which is consistent with the thermal relaxation of the high-resistive state observed in Figure 2b.An additional effect of the increased temperature is enhanced scattering, which may also cause departure from the semiclassical transport regime and the associated linear magnetoresistance.

Defect engineering:
Next, we can consider three ways of tuning the magnetoresistance through defect engineering, namely by using different growth conditions, post-growth annealing in oxygen and sample aging (Figure 4b-c).

(I) Varying the growth conditions:
The conductivity in the γ-Al2O3/SrTiO3 heterostructures results from oxygen vacancies formed during the deposition of γ-Al2O3 on SrTiO3.By controlling the oxygen partial pressure during the pulsed laser deposition of γ-Al2O3 on SrTiO3, the amount of oxygen vacancy defects in SrTiO3 can be controlled, which directly links to the resulting sheet carrier density.Therefore, Figure 4b-c effectively constitutes a defect-property relationship chart where the growth-induced defects determine the resulting mobility and magnetoresistance.

(II) Sample aging:
A boost in the electron mobility at the γ-Al2O3/SrTiO3 heterostructure after sample aging at room temperature was previously studied numerically and experimentally (2,10).Numerically, the oxygen vacancy donors were found to localize at the γ-Al2O3/SrTiO3 heterointerface due to broken lattice symmetry.In addition, an oxygen vacancy front was found to gradually diffusive deeper into the bulk of SrTiO3 (10) as schematically illustrated by the oxygen vacancy profile in the right panel of Figure S12.For convenience, the simulated results from Ref. (10) are shown in Figure S13.The resulting electrostatic potential from the positively charged oxygen vacancies caused the electrons to also shift gradually deeper into SrTiO3 while being primarily located in the region between the interface and the diffusion front (Figure S13a).This sub-interface region formed an 'electron highway' comprising a high concentration of electrons and a low concentration of oxygen vacancy scattering sites after several months of room temperature storage.The selfenhancing high mobility resulting from this dynamic donor-electron separation (Figure S13b) serves to further promote the magnetoresistance through the positive correlation between the mobility and magnetoresistance (Figure 4f).In addition, the aging is also predicted to widen the depth distribution of the electron gas, effectively increasing the out-of-plane motion of the electrons that is a prerequisites for the guiding center model (14).(III) Post-growth annealing: A detrimental effect is observed after annealing at 200 ᵒC in oxygen where the magnetoresistance is strongly reduced (Figure 3b).We attribute this decrease in magnetoresistance to a lowering of the electron mobility (Figure S8) which concurs with an increase in the effective disorder as justified below.In a recent study, scanning SQUID measurements conducted on disordered LaAlO3/SrTiO3 heterostructures were used to track the low-temperature current distribution after consecutive post-growth annealing steps in oxygen (26), similar to those applied in Figure 3b.As the conductivity originates from thermodynamically unstable oxygen vacancies in both γ-Al2O3/SrTiO3 and disordered LaAlO3/SrTiO3 (22,24), the annealing reduced the itinerant carrier density in both material systems by annihilation of oxygen vacancies.Across the consecutive annealing steps, the current distribution measured using the scanning SQUID measurements transitioned from weakly modulated around current holes to heavily modulated around both previous and newly generated current holes (26).Eventually, the current showed a strongly disordered filament-like current flow close to the metal/insulator transition induced by the annealing process.This behavior was attributed primarily to a lowering of the Fermi energy level with respect to the disorder potential (26).Interestingly, according to the guiding center model, the Hall angle in 3D Dirac metals can be tuned by varying the Fermi energy level with respect to the characteristic disorder potential variations as tan() ∝ (  / 0 ) 3/2 (14).By comparison with our present result (Figure S8d), we find that |tan()| drops by a factor of 2 in γ-Al2O3/SrTiO3 upon annealing at conditions found to lower the Fermi energy level with respect to the disorder potential disordered LaAlO3/SrTiO3.This establishes annealing and the associated defect engineering as a knob to dynamically control the effective degree of disorder and the XMR performance.By comparison with the scanning SQUID study, the evolution during the annealing also suggests that the disorder does not arise from isolated and thermodynamically unstable oxygen vacancies but may rather be caused by oxygen vacancy clusters or other extended defects.This is consistent with the large characteristic length of the disorder potential.

Figure S5 (
Figure S5 (top figures): Sheet resistance (Rs) as a function of temperature displayed for various temperatures (a) prior and (b) after room temperature storage for 239 days.(c) The associated activation barriers (Ea) before and after annealing extracted as in Figure S3.

Figure S6 (
Figure S6 (right figures): Temperature dependent (a) sheet resistance (Rs), (b) sheet carrier density (ns) and (c) zero-field electron mobility (µ) prior and after room temperature storage for 239 days.The sheet carrier density and electron mobility are extracted from the linear Hall slope around zero magnetic field.Similar trends are seen in other γ-Al2O3/SrTiO3 heterostructures as shown elsewhere (2).

Figure S7 :
Figure S7: Sheet resistance (Rs) as a function of temperature displayed for various temperatures (a) prior to annealing and (b-d) after three consecutive annealing steps at 200 ᵒC in oxygen.

Figure S8 :
Figure S8: Temperature dependent (a) sheet resistance (Rs), (b) sheet carrier density (ns) and (c) electron mobility (µ) prior to and after the annealing steps described in Figure S7.The sheet carrier density and electron mobility are extracted from the linear Hall slope around zero magnetic field.(d) Tangent to the Hall angle as a function of the magnetic field for the various annealing steps.The data in Figure a-c have also been displayed elsewhere by the authors (3).

Supplementary Section 7 :Figure S9 :
Figure S9: The magnetic flux from an alternating current detected using a scanning superconducting quantum interference device (SQUID).(ac) Several scanning SQUID scans have been merged to produce a large-area view of the local current inhomogeneities in three γ-Al2O3/SrTiO3 heterostructures with varying sheet carrier densities (ns).In panel c, we removed large-scale variations from the data by a high pass filter to get a better view of the local changes in the magnetic field distribution.The large-scale signals indicate the overall direction of the current flow.

Figure S10 :
Figure S10: Magnetoresistance (MR) at 15 T as a function of the zero-field mobility (µ0) presented for a range of samples with variations in the deposition parameters or post-processing through annealing and aging as described in the main text.Fitting the magnetoresistance as a function of the mobility reveals that  ∝  0 1.4±0.17for the compiled data with  0 < 20,000 cm 2 /Vs.For data with higher mobility, the trend remains positive but with an imprecise scaling relationship as the data is more scattered in this region.The scattered points originate from sample-to-sample variations as well as from the inclusion of data with   > 8 ⋅ 10 14 cm -2 where bulk conductivity is likely to start emerging.

Figure S11 :
FigureS11: Kohler scaling on a (a) linear plot and (b) double-logarithmic plot showing an overall reasonable collapse of the field dependent sheet resistance (Rs) normalized with the zero-field sheet resistance when these are plotted against the ratio B/Rs.Despite collapsing the curves to a fair extent, the scaling is not perfect.In particular, deviations of Kohler's rule were previously found at temperatures below 40 K for γ-Al2O3/SrTiO3, which was attributed to interactions between itinerant electrons and a magnetic order.This interaction was inferred to cause a negative magnetoresistance contribution to be added to an overall positive magnetoresistance of γ-Al2O3/SrTiO3 (4).

Figure S12 :
Figure S12: Schematics of the proposed mechanism.Schematic illustrations of the semiclassical magnetotransport in high perpendicular magnetic fields when high-mobility electrons (a) are strictly confined in 2D, (b) are restricted to move vertically by a disorder potential in some locations (red vertical line) but allowed in other locations (black vertical lines) and (c) are heated to reduce the mean-free path with the thermal energy exceeding the disorder potential variations, which leads to unrestricted vertical movement.

Figure S13 :
Figure S13: (a) Simulated depth profile of oxygen vacancies (top panel), itinerant electrons (middle panel) and electrostatic potential (lower panel) in γ-Al2O3/SrTiO3 as a function of aging time at room temperature.Here, x denotes the distance from the interface.(b) Electron mobility at 2 K (µT=2K) as a function of room temperature aging time (t).The black markers correspond to simulations whereas the colored markers are experimental values.The sample shown here is different from that presented in Figure 3 and Figure S5, although both are γ-Al2O3/SrTiO3 deposited in the same PLD chamber by the same person.The figure is adapted from Ref. (10) and reproduced with the present aesthetic modifications performed in Ref. (25).